A History of Mechanics by René Dugas

Author:René Dugas [Dugas, René]
Language: eng
Format: epub
ISBN: 9780486173375
Publisher: Dover Publications
Published: 2012-10-10T16:00:00+00:00

where v + dv is the velocity of the body at the time t + dt (if the motion is continuous). In impact, the same moment of activity would be written

where Δv is a finite increment.

Carnot next introduces the force of inertia by means of the following definition—“The resistance offered by a body to a change of state ” or the “reactions opposed to a system of bodies which make it pass from rest to motion. ” For example, in an impact (the external actions being supposed negligable) the force of inertia of a body of mass m whose velocity changes from to would be, in Carnot’s sense, m ( − ). Here the force of inertia coincides with the quantity of motion lost. But, in general, the quantity of motion lost is the “resultant of the quantity of motion produced by the motive force and the quantity of motion produced by the force of inertia. ” Finally, Carnot understands the force exerted on a body of the system to be the resultant of the motive force and the force of inertia.

In passing, we note a curious discussion on this subject. In his Sixty-Sixth Letter to a German Princess, Euler had criticised the expression “force of inertia” as uniting the concept of force (capable of changing the state of a body) and the word inertia (expressing the property of a body that tends to preserve it in its state).

Carnot objected that “the inertia is merely a property which may not be introduced in the calculations, while the force of inertia is a real measurable property ; it is the quantity of motion, which this body imparts to any other body, that displaces it from its state. ” 250

Carnot assumed the following postulates as a foundation for his mechanics.

1) The principle of inertia.

2) A system in equilibrium remains in equilibrium under the application of forces which are in equilibrium among themselves.

3) In a system of forces in equilibrium, each force is equal and opposed to the geometric sum of all the others.

4) “The quantities of motion of motive forces which, in a system of bodies, destroy each other at all times, can always be decomposed into other forces which are, taken in pairs, equal and directly opposed along the direction of the straight line which connects the two bodies to which they belong. And, in each of these bodies, each force can be regarded as nullified by the action of the other.

5) The action of one body on another by impact, traction or pressure, only depends on the relative velocity of the bodies.

6) “The quantities of motion or the dead forces which the bodies impress on each other through threads or rods are directed along these threads or rods ; and those which they impress on each other by impact or pressure are directed along the perpendicular erected at their common surface at the point of contact. ”

7) Hypotheses expressing the laws of inelastic, elastic and partially-elastic impact.

Given these definitions, Carnot introduced the concept of geometrical motion into mechanics in the following way.

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